We end this section with the following remark. Despite the asymptotic properties for \(\hat{\beta }_{\mathrm {AL}}\)  and \(\hat{\beta }_{\mathrm {ALDS}}\)  and the conditions under which they are valid, are (almost) identical, the proofs are completely different and rely upon different tools. For \(\hat{\beta }_{\mathrm {AL}}\) , our contribution is to extend the proof by [1]} who consider only the increasing domain framework (i.e. \(\mu _n=O(|D_n|)\)  and \(D_n\rightarrow \mathbb {R}^d\)  as \(n\rightarrow \infty \) ), \(s_n=s\)  and \(p_n=p)\) . The results for \(\hat{\beta }_{\mathrm {ALDS}}\)  are the first ones available for spatial point processes. To handle this estimator, we first have to study existence and optimal solutions for the primal and dual problems.
